On Cohen’s proof of the Factorization Theorem

POLONICI MATHEMATICI LXXV.2 (2000)

On Cohen’s proof of the Factorization Theorem

by Jan Kisy´ nski (Lublin)

Dedicated to the memory of Mieczys law Altman

Abstract. Various proofs of the Factorization Theorem for representations of Banach algebras are compared with its original proof due to P. Cohen.

Various proofs of the Cohen–Hewitt Factorization Theorem for represen- tations of Banach algebras are presented and discussed in [H–R;II], pp. 263–

290, [D–W], pp. 93–106 and 248–251, and [Pal], pp. 534–538.

The purpose of the present paper is to stress that all these proofs, and also some others, essentially rely upon a lemma which is implicitly con- tained in Cohen’s paper [C]. This lemma appears below as Lemma 1 of Section 2.

Our discussion of various proofs of the Cohen–Hewitt Factorization The- orem is contained in Sections 2 and 3.2–3.4. Section 3.1 concerns the earlier factorization results of R. Salem, A. Zygmund and W. Rudin. In Sections 3.5–3.7 some papers are mentioned containing either more elaborate fac- torization theorems of the Cohen–Hewitt type or factorization results of another nature.

Acknowledgements. The author is greatly indebted to Wojciech Choj- nacki for stimulating discussions and bibliographical hints.

1. The Cohen–Hewitt Factorization Theorem. The left approxi- mate identity in a Banach algebra A is, by definition, a net (e ι ) ι∈I ⊂ A such that

(1) lim

ι ke ι a − ak A = 0 for every a ∈ A.

2000 Mathematics Subject Classification: Primary 46H25; Secondary 43A20.

Key words and phrases : factorization for a representation of a Banach algebra with a bounded approximate identity.

[177]

A left approximate identity (e _ι ) _ι∈I ⊂ A is said to be bounded if sup

ι∈I

ke ι k A < ∞.

If A is a Banach algebra with bounded left approximate identity (e ι ) ι∈I and T is a continuous representation of A on a Banach space X, then

(2) lim

ι kT (e ι )y − yk X = 0 for every y ∈ span T (A)X.

This follows from the fact that (T (e ι )) ι∈I is a bounded net in L(X) such that lim _ι kT (e ι )y − yk = 0 for every y ∈ T (A)X ( ¹ ).

The Factorization Theorem (P. Cohen [C], 1959; E. Hewitt [H], 1964). If A is a Banach algebra with bounded left approximate identity (e _ι ) _ι∈I and T is a continuous representation of A on a Banach space X, then T (A)X is a closed subspace of X. Furthermore, for every y ∈ T (A)X and every ε > 0 there are a ∈ A and x ∈ T (A)y such that T (a)x = y, kx − yk ≤ ε, and

(3) a = X ∞ n=1

p n e ι n where ι n ∈ I, p n > 0 for n = 1, 2, . . . and X ∞ n=1

p n = 1.

The assertions of the Factorization Theorem may be expressed by the single condition

(4) for every y ∈ span T (A)X and every ε > 0 there are a ∈ A and x ∈ T (A)y such that T (a)x = y, kx − yk ≤ ε and (3) is satisfied.

2. Cohen’s proof of the Factorization Theorem. Let A be a Banach algebra, and T a continuous representation of A on a Banach space X. Sup- pose that there is a bounded left approximate identity in A. Then Cohen’s proof of the existence of a factorization

y = T (a)x, a ∈ A, x ∈ X,

of an element y of span T (A)X consists in approximation by a sequence of factorizations

y = e T (a _n )x _n , a _n ∈ G, x n = e T (a ⁻¹ _n )y, n = 1, 2, . . . ,

where G is the set of invertible elements of the unitization A u of A, and e T is a representation of A u on X extending the representation T .

( ¹ ) From (2) it follows at once that span T (A)X = T (A)X, i.e. T (A)X is a closed

linear subspace of X. The factorization theorem says more: T (A)X itself is a closed linear

subspace. This last is almost trivial when A contains a unit e. Indeed, since T (e) =

T (e · e) = T (e)T (e), it follows that T (e) is a projector and hence T (e)X is a closed

subspace. Furthermore, T (A)X = T (eA)X = T (e)T (A)X ⊂ T (e)X ⊂ T (A)X, so that

T (A)X = T (e)X.

The unitization A u of A is the Banach algebra with unit such that 1 ^o as a linear space, A u is equal to the direct sum K + A, where K is the field of scalars of A,

2 ^o A u = K + A is equipped with the norm k k A u such that kλ + ak A u =

|λ| + kak A for every λ ∈ K and a ∈ A, 3 ^o the multiplication in A u is defined by

(λ + a)(µ + b) = λµ + (µa + λb + ab), where λ, µ ∈ K and a, b ∈ A.

The unit in A u is 1 = 1 + 0 ∈ K + A ( ² ). Let ϕ : A u → K, π : A u → A

be the projectors corresponding to the splitting A u = K + A. From 3 ^o it fol- lows that ϕ is a multiplicative functional on A u . Let e T be the representation of A _u on X such that

T (a) = ϕ(a) + T (πa) e for every a ∈ A u .

The following lemma is implicitly contained in Cohen’s paper [C].

Lemma 1. Under the assumptions of the Factorization Theorem, let E = {e ι : ι ∈ I}, M = sup

e∈E

kek A .

Take any y ∈ span T (A)X, a ∈ G, γ ∈ (0, 1/(M + 1)), δ > 0. Then there exists an ea ∈ G satisfying the three conditions:

ϕ(ea) = ϕ(a) − γϕ(a), (i)

πea = πa + γϕ(a)e for some e ∈ E, (ii)

k e T (ea ⁻¹ )y − e T (a ⁻¹ )yk X ≤ δ.

(iii)

Note that (i)&(ii) may be written as one condition

(iv) ea = a + γϕ(a)(e − 1),

and that (i)&(ii) implies

(v) kπea − πak A ≤ γM |ϕ(a)| = M |ϕ(ea) − ϕ(a)|.

Lemma 1 implies the Factorization Theorem. Let p _n = λ _n−1 − λ n for n = 1, 2, . . . ,

where λ 0 , λ 1 , . . . is a positive strictly decreasing sequence such that

(5)

λ 0 = 1, lim

n→∞ λ _n = 0, 1 − λ n

λ n−1

= γ n ∈

 0, 1

M + 1



for n = 1, 2, . . .

( ² ) If A has a unit e, then A u makes sense, but e is no longer a unit in A u .

Then (6)

λ _n = (1 − γ _n )λ _n−1 ,

p _n = γ _n λ n−1 > 0 for n = 1, 2, . . . , and X ∞ n=1

p _n = 1.

The concrete form of the sequence λ ₀ , λ ₁ , . . . is inessential. One can follow [C] and take λ _n = (1 − γ) ⁿ , γ = const ∈ (0, 1/(M + 1)), so that γ _n = γ, or λ _n = (1 + M )/(1 + M + n), so that γ _n = 1/(1 + M + n). From (5), (6) and (i)&(v)&(iii) it follows that for every y ∈ span T (A)X and every ε > 0 there exists a sequence a 0 , a 1 , . . . of elements of G such that a 0 = 1 and

ϕ(a n ) = λ n , (i) 0

kπa n − πa n−1 k A ≤ M p n , (v) 0

k e T (a ⁻¹ _n )y − e T (a ⁻¹ _n−1 )yk X ≤ εp n

(iii) 0

for n = 1, 2, . . . From (i) 0 &(v) 0 &(iii) 0 it follows that lim _n→∞ ka n −ak A u = 0 for some a ∈ A such that

kak A = π

X ∞ n=1

(a n − a n−1 ) A ≤ M

X ∞ n=1

p n = M,

and, if x n = e T (a ⁻¹ _n )y, then lim n→∞ kx n − xk X = 0 for some x ∈ X such that

kx − yk X = kx − x 0 k X ≤ X ∞ n=1

kx n − x n−1 k X ≤ ε X ∞ n=1

p n = ε.

Since y = e T (a n )x n for every n = 0, 1, . . . , a passage to the limit implies that y = e T (a)x = T (a)x. By (2), y = lim _ι T (e _ι )y ∈ T (A)y, whence

x n = e T (a ⁻¹ _n )y = ϕ(a ⁻¹ _n )y + T (πa ⁻¹ _n )y ∈ T (A)y

for n = 0, 1, . . . , and so x = lim _n→∞ x _n ∈ T (A)y. Thus (i)&(v)&(iii) implies (4) _M for every y ∈ span T (A)X and every ε > 0 there are a ∈ A and

x ∈ T (A)y such that kak A ≤ M , kx − yk X ≤ ε and T (a)x = y.

The statement (4) M is a weakened version of (4): the condition (3) is replaced by a ∈ A, kak _A ≤ M . In order to prove (4), instead of (i)&(v)&(iii) one has to use (iv)&(iii). Indeed, (iv)&(iii) implies that for every y ∈ span T (A)X and every ε > 0 there exists a sequence a ₀ , a ₁ , . . . of elements of G such that a 0 = 1, the conditions (i) 0 and (iii) 0 are satisfied, and

(iv) 0 for every n = 1, 2, . . . there is e _n ∈ E such that

a _n = a n−1 + γ _n ϕ(a n−1 )(e _n − 1) = a n−1 + p _n (e _n − 1).

Since (iv) 0 implies (v) 0 , the preceding argument leading to (4) M remains in

force. Furthermore, from (iv) 0 it follows that a _n = λ _n + p 1 e 1 + . . . + p _n e _n for every n = 1, 2, . . . and hence, by (5) and (6), lim n→∞ ka n − ak A u = 0, where a = P ∞

n=1 p _n e _n .

Proof of Lemma 1 ( ³ ). For every ι ∈ I define ea ι = a + b ι , b ι = γϕ(a)(e ι − 1).

Lemma 1 will follow once it is shown that

(I) there is ι 0 ∈ I such that ea ι ∈ G whenever ι 0 ≺ ι, and

(II) lim

ι k e T (ea ⁻¹ ι )y − e T (a ⁻¹ )yk X = 0.

In the proof of (I) we will use the equality ea ι = (1 + b ι a ⁻¹ )a.

Notice that

b ι a ⁻¹ = γϕ(a)(e ι − 1)[ϕ(a ⁻¹ ) + πa ⁻¹ ]

= γ(e _ι − 1) + γϕ(a)(e ι − 1)πa ⁻¹ , whence

(7) kb ι a ⁻¹ k A u ≤ γ(M + 1) + γ|ϕ(a)| · k(e ι − 1)πa ⁻¹ k A . Since γ ∈ (0, 1/(M + 1)), one can choose θ such that

γ(M + 1) < θ < 1.

By (1) and (7), there is ι ₀ ∈ I such that

kb ι a ⁻¹ k A u ≤ θ whenever ι ₀ ≺ ι.

An application of the C. Neumann series shows that then 1 + b ι a ⁻¹ ∈ G and k(1 + b _ι a ⁻¹ ) ⁻¹ k A u ≤ (1 − θ) ⁻¹ . In consequence, whenever ι ₀ ≺ ι, then ea ι = (1 + b _ι a ⁻¹ )a ∈ G ( ⁴ ) and

(8) kea ⁻¹ ι k A u ≤ ka ⁻¹ k A u (1 − θ) ⁻¹ . For the proof of (II) observe that if ι ₀ ≺ ι, then

ea ⁻¹ ι − a ⁻¹ = ea ⁻¹ ι (a − ea ^ι )a ⁻¹ = −ea ⁻¹ ι b ι a ⁻¹ = −γϕ(a)ea ⁻¹ ι (e ι − 1)a ⁻¹ . As a consequence, by (8),

k e T (ea ⁻¹ ι )y − e T (a ⁻¹ )yk _X

≤ γ|ϕ(a)| · k e T k L(A u ;L(X)) ka ⁻¹ k A u (1 − θ) ⁻¹ k[T (e ι ) − 1] e T (a ⁻¹ )yk X .

( ³ ) This proof differs from the argument in [C]; see Section 3.2.

( ⁴ ) This proof of the invertibility of e a ι follows [P;3], pp. 283–284, and [P–P], p. 136.

The elements b, b ^′ and c of A u appearing in [P;3] correspond to our a ⁻¹ , e a ⁻¹ ι and (1 +

b ι a ⁻¹ ) ⁻¹ . In [P–P] the elements of A u are written in the form a + r, a ∈ A, r ∈ K, so

that our πa ⁻¹ corresponds to (a + r) ⁻¹ − r ⁻¹ in [P–P].

From this estimate and from (2) the equality (II) follows, because e T (a ⁻¹ )y = ϕ(a ⁻¹ )y + T (πa ⁻¹ )y ∈ span T (A)X.

3. Forerunners of the Cohen–Hewitt Factorization Theorem, variants of the proof, and references to other factorization results 3.1. Forerunners of the Cohen–Hewitt theorem. Let G be a locally compact group with a fixed left-invariant Haar measure µ, and let L ¹ (G) be the space of the equivalence classes of functions on G integrable with respect to µ. The convolution a ∗ b of two elements, a and b, of L ¹ (G) is defined by

(a ∗ b)(g) =

\

G

a(h)b(h ⁻¹ g) µ(dh) for a.e. g ∈ G.

Then L ¹ (G) is a convolution Banach algebra. Cohen’s paper [C] contains the following results which follow from the Factorization Theorem applied to A = L ¹ (G), X = L ¹ (G), or X = C(G) if G is compact, and to the representation of A on X defined by T (a)x = a ∗ x:

(a) L ¹ (G) = L ¹ (G) ∗ L ¹ (G) for every locally compact group G. This was proved earlier in particular cases: for G = T, the circle group, by R. Salem and A. Zygmund, and for G = R and G a locally euclidian abelian group by W. Rudin [R;1], [R;2].

(b) C(G) = L ¹ (G) ∗ C(G) for every compact group G. In the particular case of G = T this was proved earlier by R. Salem and A. Zygmund.

(c) Strictly positive factorization: if G is a compact group, y ∈ C(G) and inf _G y > 0, then there are a ∈ L ¹ (G) and x ∈ C(G) such that ess inf _G a > 0, inf G x > 0 and a ∗ x = y. In the deduction of this result from the Fac- torization Theorem the conditions (3) and kx − yk ≤ ε are essential. Note that Cohen [C] proved that an analogous non-negative factorization does not hold.

In their proofs of the particular cases of (a) and (b), R. Salem, A. Zyg- mund and W. Rudin used the Fourier methods which are not applicable in the general situation. Nevertheless it seems interesting to compare the condition (3) from the factorization theorem with the formulas labelled be- low by (∗) and (∗ ∗∗), appearing in the proof of the factorization L ¹ (R) = L ¹ (R) ∗ L ¹ (R) presented in [R;2]. The approximate identity in L ¹ (R) ap- pears there in the from e _t = K _t for t ∈ (0, ∞), where

K _t (x) = t 2π

 sin(tx/2) tx/2

 2

is the Fej´er kernel. The Fourier transform of K _t is

K b t (ξ) = (1 − |ξ|/t) ⁺ .

For given f ∈ L ¹ (R) Rudin proves that f = a ∗ g where a ∈ L ¹ (R) and g ∈ L ¹ (R) are defined as the integrals of continuous L ¹ (R)-valued functions

a =

∞

\

0

tλ ^′′ (t)K _t dt, (∗)

g = f +

∞

\

0

α(t)(f − K t ∗ f ) dt.

(∗ ∗)

Here α ∈ C[0, ∞) and λ ∈ C ² [0, ∞) are positive functions such that

∞

\

0

α(t) dt = ∞,

∞

\

0

α(t)



kK t ∗ f − f k + 1 t



dt < ∞, λ(t) = 1 Φ(t) , where

Φ(t) = 1 +

t

\

0

α(s) ds + t

∞

\

t

α(s) s ds.

These definitions imply that g ∈ L ¹ (R), λ ^′′ is positive and (∗ ∗∗)

∞

\

0

tλ ^′′ (t) dt = 1,

so that a ∈ L ¹ (R), and furthermore, ba(ξ) = λ(|ξ|) and bg(ξ) = Φ(|ξ|) b f (ξ) ( ⁵ ), whence babg = b f , and so a ∗ g = f .

Discrete forerunners of (∗) and (∗ ∗∗) appear in the book of A. Zygmund [Z;I] as (1.7) on p. 183, (4.2) on p. 93, and Theorem 1.5 on p. 183 (attributed on p. 378 to H. W. Young (1913) and A. N. Kolmogorov (1923)). A discrete forerunner of (∗ ∗) is the second formula in line 111 ³ of R. Salem’s paper [S], related to a limit passage in (3) on p. 110 of [S]. A. Zygmund [Z;I], p. 378 11-6 , points out that the theorems from [S] together with Theorem 1.5 of Chap. V of [Z;I], p. 183, lead to the factorization results for G = T mentioned in (a) and (b). A discrete analogue of the whole elegant Rudin construction is used in Sec. 7.5 of the book of R. E. Edwards [E;I] for reproving the Salem–

Zygmund results. Remarks on the Fourier factorization may be found in Sec. 3.1.1(c) and 7.5.2–4 of [E;I] and Sec. 11.4.18(6) of [E;II].

( ⁵ ) Indeed, Φ ^′ (t) =

T

∞

t (α(s)/s) ds, Φ ^′′ (t) = −α(t)/t ≤ 0, λ ^′ (t) = −Φ ^′ (t)/[Φ(t)] ² ≤ 0, λ ^′′ (t) = (2[Φ ^′ (t)] ² − Φ ^′′ (t)Φ(t))[Φ(t)] ⁻³ ≥ 0. Hence 0 ≤ −tλ ^′ (t) = tΦ ^′ (t)/[Φ(t)] ² ≤ [Φ(t)] ⁻¹ , so lim t→∞ tλ ^′ (t) = lim t→∞ λ(t) = 0 and

T

∞

0 tλ ^′′ (t) dt = tλ ^′ (t)| ^t=∞ _t=0 −

T

∞ 0 λ ^′ (t) dt

= λ(0) = 1. Moreover, (∗) implies b a(ξ) =

T

∞

0 tλ ^′′ (t)(1 − |ξ|/t) ⁺ dt =

T

∞

|ξ| (t − |ξ|)λ ^′′ (t) dt = (t − |ξ|)λ ^′ (t)| ^t=∞ _t=|ξ| −

T

∞

|ξ| λ ^′ (t) dt = λ(|ξ|), and bg(ξ) = [1 +

T

∞

0 α(t)(1 − b K t (ξ)) dt] b f (ξ) =

Φ(|ξ|) b f (|ξ|) by (∗ ∗).

3.2. The original calculations of Cohen. As already mentioned, Lemma 1 is implicitly contained in Cohen’s paper [C]. The argument used in [C] for proving this implicit lemma was repeated in [V], [H], [C–FT], [J], [G–L–R], [T], [S–T], [H–R;II], [B–D], [A–S], [D–W], [G] and [Pal]. In our notation, this argument is as follows. For every ι ∈ I one has kγ(e ι −1)k ≤ γ(M +1) = ϑ < 1, whence 1 − γ + γe _ι ∈ G and k(1 − γ + γe ι ) ⁻¹ k ≤ (1 − ϑ) ⁻¹ . The possibility of choosing ι 0 ∈ I so that

(I) ea ι = a + γϕ(a)(e _ι − 1) ∈ G whenever ι 0 ≺ ι is proved in [C] by means of the formulas

(9) ea ι = (1 − γ + γe ι )a ι , a ι = ϕ(a) + (1 − γ + γe ι ) ⁻¹ πa.

Since

ka ι − ak = k(1 − γ + γe ι ) ⁻¹ πa − πak

= kγ(1 − γ + γe ι ) ⁻¹ (1 − e ι )πak

≤ γ(1 − ϑ) ⁻¹ ke ι πa − πak, it follows by (1) that

(10) lim

ι ka ι − ak = 0.

Hence there is ι 0 ∈ I such that if ι 0 ≺ ι, then ka ι − ak ≤ ¹ ₂ ka ⁻¹ k ⁻¹ , and so a ι = [1 + (a ι − a)a ⁻¹ ]a ∈ G and ka ⁻¹ _ι k ≤ 2ka ⁻¹ k. In consequence, if ι 0 ≺ ι, then ea ι = (1 − γ + γe _ι )a _ι ∈ G, proving (I). The equality

(II) lim

ι k e T (ea ⁻¹ ι )y − e T (a ⁻¹ )yk = 0 is proved in [C] as follows. If ι 0 ≺ ι, then

ea ⁻¹ ι − a ⁻¹ = (a ⁻¹ _ι − a ⁻¹ )(1 − γ + γe _ι ) ⁻¹ + a ⁻¹ [(1 − γ + γe _ι ) ⁻¹ − 1]

= − a ⁻¹ _ι (a ι − a)a ⁻¹ (1 − γ + γe ι ) ⁻¹

− γa ⁻¹ (1 − γ + γe _ι ) ⁻¹ (e _ι − 1), whence

k e T (ea ⁻¹ ι )y − e T (a ⁻¹ )yk

≤ k e T k · ka ⁻¹ k(1 − ϑ) ⁻¹ {2ka ⁻¹ k · kyk · ka ι − ak + γkT (e ι )y − yk}.

This estimate implies (II), by (10) and (2).

3.3. Variants of Cohen’s proof. The method of Koosis. The reasoning presented in Section 3.2 becomes simpler when the elements ea ι = a + γϕ(a)(e ι − 1) satisfying (9) are replaced by

ba ι = (1 − γ + γe _ι )a.

Then, for every ι ∈ I, ϕ(ba ^ι ) = (1 − γ)ϕ(a),

kπba ι − πak = kγϕ(a)e ι + γ(e _ι − 1)πak ≤ γ|ϕ(a)|M + γke ι πa − πak, ba ι ∈ G, and ba ⁻¹ ι − a ⁻¹ = −γa ⁻¹ (1 − γ + γe _ι ) ⁻¹ (e _ι − 1), whence

k e T (ba ⁻¹ ι )y− e T (a ⁻¹ )yk ≤ γk e T k·ka ⁻¹ k(1−ϑ) ⁻¹ kT (e ι )y−yk, ϑ = γ(M +1).

This yields

Lemma 2. Under the notation of Sections 1 and 2, and the assump- tions of the Factorization Theorem, let y ∈ span T (A)X, a ∈ G, γ ∈ (0, 1/(M + 1)), δ > 0. Then there is ba ∈ G such that ba = (1 − γ + γe)a for some e ∈ E,

kπba − πak ≤ γ|ϕ(a)|M + δ and k e T (ba ⁻¹ )y − e T (a ⁻¹ )yk ≤ δ.

For given y ∈ span T (A)X, ε > 0 and γ ∈ (0, 1/(M +1)), using Lemma 2, one can define inductively e 1 , e 2 , . . . in E so that the formulas

(11) a 0 = 1, a _n = (1 − γ + γe _n )a n−1 for n = 1, 2, . . .

define a sequence a 0 , a 1 , . . . in G such that ϕ(a n ) = (1−γ) ⁿ , kπa n −πa n−1 k ≤ γ(1−γ) ⁿ⁻¹ (M +ε) and k e T (a ⁻¹ _n )y− e T (a ⁻¹ _n−1 )yk ≤ γ(1−γ) ⁿ⁻¹ ε for n = 1, 2, . . . This yields

(b 4) _M for every y ∈ span T (A)X and every ε > 0 there are ba ∈ A and b x ∈ T (A)y such that kbak ≤ M + ε, kb x − yk ≤ ε and T (ba)b x = y.

If, for given y ∈ span T (A)X and ε > 0, ba and b x satisfy (b 4) M , then for a = _M ^M _+ε ba and x = ^M _M ^+ε x one has kak ≤ M and b

kx−yk ≤ kb x−yk+ ε

M kb xk ≤ kb x−yk+ ε

M kyk+ ε

M kb x−yk ≤



1+ kyk + ε M

 ε.

Hence (b 4) _M is equivalent to (4) _M from Section 2.

It was P. Koosis [K] who showed that the elements a ∈ A and x ∈ T (A)y satisfying T (a)x = y for a given y ∈ span T (A)X may be determined by an approximation a = lim a _n , x = lim e T (a ⁻¹ _n )y, where a 0 = 1 ∈ A u , a n = (1 − γ + γe n )a n−1 for n = 1, 2, . . . , γ = const ∈ (0, 1/(M + 1)), and e ₁ , e ₂ , . . . is a suitable sequence of elements of E. This idea was continued by M. Altman [A;1–5], R. S. Doran and J. Wichmann [D–W], pp. 97–100, J. Esterle [Es;4] and H. G. Feichtinger and M. Leinert [F–L], still for Banach algebras.

W. ˙Zelazko [ ˙Z], Sec. 6.4, pp. 24–26, showed that the Koosis–Altman formulas (11) may be replaced by

(12) a 0 = 1, a n = Φ γ (e n )a n−1 for n = 1, 2, . . . , where γ > 0 is a constant such that γ(M − 1) < 1, and

Φ γ (e) = (1 + γ − γe) ⁻¹

for every e ∈ E. Then

kΦ γ (e)k ≤ (1 + γ − γM ) ⁻¹ , Φ _γ (e) − 1 = γΦ _γ (e)(e − 1), [Φ _γ (e)] ⁻¹ − 1 = −γ(e − 1),

which implies

Lemma 3. Under the notation of Sections 1 and 2, and the assump- tions of the Factorization Theorem, let y ∈ span T (A)X, a ∈ G, γ > 0, γ(M − 1) < 1, δ > 0. Then there is e ∈ E such that if b ba = Φ γ (e)a, then bba ∈ G,

kπb ba − πak = kbba − ak − |ϕ(bba) − ϕ(a)| ≤ γ(1 + M )

1 + γ − γM |ϕ(a)| + δ − γ

1 + γ |ϕ(a)|

and

k e T (b ba ⁻¹ )y − e T (a)yk ≤ δ.

For given y ∈ span T (A)X, ε > 0 and γ > 0 such that γ(M − 1) < 1, using Lemma 3, one can define inductively e 1 , e 2 , . . . in E so that the sequence a 0 , a 1 , . . . of elements of G defined by (12) satisfies

ϕ(a n ) = (1 + γ) ⁻ⁿ , kπa n − πa n−1 k ≤

 γ(1 + M )

1 + γ − γM − γ 1 + γ



(1 + γ) ¹⁻ⁿ + ε 2 ⁿ⁺¹ , and k e T (a ⁻¹ _n )y − e T (a ⁻¹ _n−1 )yk ≤ ε/2 ⁿ . If a = lim a _n and x = lim e T (a ⁻¹ _n )y, then a ∈ A, x ∈ T (A)y, y = T (a)x, kx − yk ≤ ε and

kak ≤ X ∞ n=1

kπa n − πa n−1 k ≤ (1 + γ)(1 + M )

1 + γ − γM − 1 + ε 2 . For γ > 0 such that

γ(M − 1) < 1 and (1 + γ)(1 + M )

1 + γ − γM ≤ 1 + M + ε 2 , one obtains (b 4) _M , which is equivalent to (4) _M .

In [H–R;II], Sec. 32.50, pp. 287–288, an argument similar to that of

˙Zelazko is carried out under the assumptions that M = 1 and the approxi- mate identity is two-sided. Instead of (12) Hewitt and Ross use the formula a n = a n−1 ϕ(e n ) where ϕ(e) = ¹ ₃ 

1 + P ∞ k=1

1

3 [1 + e]
k 

is equal to our Φ 1 (e) = (2 − e) ⁻¹ = ¹ ₃ 1 − ¹ ₃ [1 + e]
−1

. A construction similar to that of

˙Zelazko also appears in [Es;1] in the proof of Theorem 2.8.

In Section 3.5 some papers will be mentioned in which the constructions

of Koosis and ˙Zelazko are applied to representations of some non-Banach

algebras. These applications are possible thanks to the fact that these con-

structions lead to sequences of factorizations y = e T (a n )x n , x n = e T (a ⁻¹ _n )y,

for which the proofs of the invertibility of a _n ’s and of the existence of lim e T (a ⁻¹ _n )y are simpler than in the case of Cohen’s construction.

3.4. Deduction of the Cohen–Hewitt Factorization Theorem from some results of V. Pt´ ak and J. Esterle. In a somewhat weakened version, namely with the condition a ∈ conv{e _ι : ι ∈ I} but without (3) ( ⁶ ), the Factor- ization Theorem may be deduced from the “induction theorem” of V. Pt´ ak ([P;3], p. 280; [P–P], p. 5; [P;1]; [P;2]), and also from the theorem of J. Es- terle about projective systems ([Es;4], p. 109, Th. 2.1). The applicability of these theorems to the factorization problem follows from Lemmas 1–3.

The proofs of both theorems are similar to the part of Cohen’s argument presented above as “Lemma 1 implies the Factorization Theorem”. Nev- ertheless, the approach to factorization via the theorem of Pt´ ak or that of Esterle is interesting because it emphasizes the geometric nature of Cohen’s construction.

Under the notation of Sections 1 and 2, consider in A u the subset E = conv({1} ∪ E) = [

0≤λ≤1

(λ + (1 − λ)conv E).

Take any y ∈ span T (A)X and ε > 0. For every λ ∈ (0, 1] let Z(λ) = {(a, x) ∈ (E ∩ G) × X :

kx − yk ≤ ε(1 − ϕ(a)), e T (a)x = y, 0 < ϕ(a) ≤ λ}.

Then (1, y) ∈ Z(1) and Z(λ ^′ ) ⊂ Z(λ ^′′ ) ⊂ (E ∩ G) × T (A)y whenever 0 <

λ ^′ < λ ^′′ ≤ 1. If (a, x) ∈ T

0<λ≤1 Z(λ), then a ∈ conv E ⊂ A, x ∈ T (A)y, kx − yk ≤ ε, and T (a)x = y. Hence the weakened version of the statement (4) will follow once it is shown that

(∗) \

0<λ≤1

Z(λ) 6= ∅.

A proof of (∗) may be obtained by using Lemma 1 and applying Pt´ ak’s or Esterle’s theorem. Indeed, Lemma 1 implies

Corollary. Fix any y ∈ span T (A)X and ε > 0. Then for every λ ∈ (0, 1], (a, x) ∈ Z(λ) and γ ∈ (0, 1/(M + 1)) there exists (ea, e x) ∈ Z((1 − γ)λ) such that ϕ(ea) = (1 − γ)ϕ(a) and

kπea − πak

M ∨ ke x − xk

ε ≤ γϕ(a) = ϕ(a) − ϕ(ea).

Define in A u × X the norm

|||(a, x)||| =

 kπak M ∨ kxk

ε



+ |ϕ(a)|, a ∈ A u , x ∈ X.

( ⁶ ) Sometimes this may be an essential loss of information, see (c) in Section 3.1.

From the Corollary it follows that for every λ ∈ (0, 1], (a, x) ∈ Z(λ) and γ ∈ (0, 1/(M + 1)) there is (ea, e x) ∈ Z((1 − γ)λ) such that |||(ea, e x) − (a, x)||| ≤ 2γλ. Since (1, y) ∈ Z(1), it follows that Z(λ) 6= ∅ for every λ ∈ (0, 1], and

(∗ ∗) sup

(a,x)∈Z(λ)

dist((a, x); Z((1 − γ)λ)) ≤ 2γλ

for every λ ∈ (0, 1] and γ ∈ (0, 1/(M + 1)), where the distance is defined by the norm ||| |||.

The implication (∗ ∗)⇒(∗) is a consequence of Pt´ak’s theorem, and also of Esterle’s. Indeed, if e Z(t) = Z(t/(2γ)) for t ∈ (0, 2γ], then (∗ ∗) takes the form

sup

(a,x)∈ e Z(t)

dist((a, x); e Z(w(t))) ≤ t for every t ∈ (0, 2γ], where w(t) = (1−γ)t. Since w(t)+(w ◦w)(t)+. . . = P ∞

n=1 (1−γ) ⁿ t < ∞ for every t ∈ (0, 2γ], the function w is a rate of convergence on (0, 2γ] ([P–P], p. 2, Def. 1). Hence the theorem of Pt´ ak ([P–P], p. 5, Proposition 1.7) implies that T

0<t≤2γ Z(t) 6= ∅, proving (∗). e

In order to see that (∗ ∗)⇒(∗) is a consequence of Esterle’s theorem ([Es;4], p. 109, Th. 2.1), take any positive strictly decreasing sequence λ 0 , λ 1 , . . . satisfying (5). Then, by (∗ ∗),

sup

(a,x)∈Z(λ _n−1 )

dist((a, x); Z(λ _n )) ≤ 2γ _n λ n−1 = 2(λ n−1 − λ n ) for n = 1, 2, . . . Since P ∞

n=1 2(λ n−1 − λ) = 2 < ∞, Esterle’s theorem implies that T ∞

n=0 Z(λ n ) 6= ∅, proving (∗).

Similar applications of the theorems of Pt´ ak and Esterle to the factoriza- tion problem are presented in [K–V]; [P–P], pp. 134–137; [D–W], pp. 100–

104; [Es;4], pp. 115–119. In [P;3] a more sophisticated factorization theorem is deduced from the “induction theorem”.

3.5. Extensions of the Cohen–Hewitt theorem to representations of some non-Banach algebras. It was observed in [O] and [Cr] that the argument of Koosis, presented in Remark 3, works in every Fr´echet algebra ( ⁷ ) with a uniformly bounded left approximate identity (u.b.l.a.i.). This idea was taken up in [W.H.S], [M.K.S], [Vo;1] and [Vo;2]. Examples of Fr´echet algebras with u.b.l.a.i. are discussed in [O] and [Vo;1]. In [D] the u.b.l.a.i. is defined for any metrizable algebra, and the argument of ˙Zelazko, described in our Sec. 3.3, is used for extending the factorization theorem to any essential continuous representation T of a complete metrizable locally convex algebra A with

( ⁷ ) That is, an algebra which is a complete metrizable l.c.v.s. and, moreover, is locally

multiplicatively convex.

u.b.l.a.i. on a complete metrizable vector space X. In [AP] this last result is generalized to complete metrizable algebras which need not be locally convex, but are fundamental ([AP], p. 54, Def. 2.1).

3.6. Simultaneous factorization, factorization involving an analytic func- tion, power factorization, and factorization by means of a one-parameter semigroup. The arguments discussed in Sections 2 and 3.1–4 were also used for proving theorems about simultaneous factorization of all the elements of a subset of span T (A)X by means of a single element of A. For the case of Banach algebras see [R;2]; [V]; [J]; [T]; [Ri]; [S–T]; [C–S];

[H–R;II], pp. 268–269, Theorem 32.23 and Corollary 32.24; [W.H.S]; [P1];

[B–D], p. 62, Corollary 12; [ ˙Z], p. 23, Theorem 6.4; [Pal], pp. 538–539, Corol- lary 5.2.3. For extensions to Fr´echet algebras (and more general algebras) see [O]; [Cr]; [M.K.S]; [ ˙Z], p. 34, Theorem C.6.1; [Vo;1]; [Vo;2]; [D], Corol- lary 4.2.

Cohen’s basic idea of imbedding a Banach algebra A into its unitiza- tion A u and using an approximation in A u also appears in [C–St] and [S;2]

in connection with the factorization y = T (f (a))x where f is an analytic function, in [A–S], [Es;1], [P;2] and [G] where the power factorization is con- sidered, and in [S;1,3,4] and [Pal], Sec. 5.3, pp. 543–548, in connection with the factorization by means of a one-parameter semigroup. The factorization theorem of A. M. Sinclair [S;4], pp. 35–37, concerns a representation T of a complex Banach algebra A on a complex Banach space X, and states that if there is a countable bounded two-sided approximate identity in A, then for every y ∈ span T (A)X there are a holomorphic map C ∋ z 7→ x _z ∈ X and a holomorphic semigroup C ⁺ ∋ z 7→ a ^z ∈ A defined on the complex right half plane C ⁺ = {z ∈ C : Re z > 0}, such that y = x ₀ = T (a ^z )x _z for every z ∈ C ⁺ , and lim _{U(Ψ )∋z→0} ka ^z b − bk _A = 0 for every b ∈ A and every sector U (Ψ ) = {z ∈ C ⁺ : |Arg z| ≤ Ψ }, 0 < Ψ < π/2. The growth properties of the holomorphic semigroups C ⁺ ∋ z 7→ a ^z ∈ A influence the structure of a Banach algebra A. See [Es;2,3], [S;4], [W], [G–W], [G–R] and other papers quoted there.

3.7. Factorization without a bounded approximate identity. Not all fac- torization theorems for topological algebras and their representations are proved by arguments of Cohen’s type, related to bounded approximate identities. Indeed, the papers [Pas], [L], [D–M], [P–V], [Vo;1] and [Ou]

deal with factorization and weak factorization in Banach algebras, and non-

Banach function algebras (with pointwise multiplication) and convolution

algebras, without bounded approximate identities. Also the proofs of weak

factorization theorems for general separable Banach algebras, presented in

[Pal], Sec. 5.3, pp. 549–552, have nothing in common with Cohen’s argu-

ment.

References

[A–S] G. R. A l l a n and A. M. S i n c l a i r, Power factorization in Banach algebras with a bounded approximate identity , Studia Math. 56 (1976), 31–38.

[A;1] M. A l t m a n, Factorisation dans les alg`ebres de Banach, C. R. Acad. Sci. Paris S´er. A 272 (1971), 1388–1389.

[A;2] —, Infinite products and factorization in Banach algebras, Boll. Un. Mat. Ital.

5 (1972), 217–229.

[A;3] —, Contracteurs dans les alg`ebres de Banach, C. R. Acad. Sci. Paris S´er. A 274 (1972), 399–400.

[A;4] —, Contractors, approximate identities and factorization in Banach algebras, Pacific J. Math. 48 (1973), 323–334.

[A;5] —, A generalization and the converse of Cohen’s factorization theorem, Duke Math. J. 42 (1975), 105–110.

[AP] E. A n s a r i - P i r i, A class of factorable topological algebras, Proc. Edinburgh Math. Soc. 33 (1990), 53–59.

[B–D] F. F. B o n s a l l and J. D u n c a n, Complete Normed Algebras, Springer, 1979.

[C] P. C o h e n, Factorization in group algebras, Duke Math. J. 26 (1959), 199–205.

[C–S] H. S. C o l l i n s and W. H. S u m m e r s, Some applications of Hewitt’s factoriza- tion theorem , Proc. Amer. Math. Soc. 21 (1969), 727–733.

[Cr] I. G. C r a w, Factorization in Fr´echet algebras, J. London Math. Soc. 44 (1969), 607–611.

[C–FT] P. C. C u r t i s, J r. and A. F i g ` a - T a l a m a n c a, Factorization theorems for Ba- nach algebras, in: Function Algebras, F. T. Birtel (ed.), Scott, Foresman and Co., Chicago, IL, 1966, 169–185.

[C–St] P. C. C u r t i s, J r. and H. S t e t k a e r, A factorization theorem for anatytic functions operating in a Banach algebra, Pacific J. Math. 37 (1971), 337–343.

[D–M] J. D i x m i e r et P. M a l l i a v i n, Factorisations de fonctions et de vecteurs ind´efi- niment diff´ erentiables, Bull. Sci. Math. (2) 102 (1978), 305–330.

[D] P. G. D i x o n, Automatic continuity of positive functionals on topological invo- lution algebras, Bull. Austral. Math. Soc. 23 (1981), 265–281.

[D–W] R. S. D o r a n and J. W i c h m a n n, Approximate Identities and Factorization in Banach Modules, Lecture Notes in Math. 768, Springer, 1979.

[E;I] R. E. E d w a r d s, Fourier Series, a Modern Introduction, Vol. I, 2nd ed., Sprin- ger, 1979.

[E;II] —, Fourier Series, a Modern Introduction, Vol. II, 2nd ed., Springer, 1982.

[Es;1] J. E s t e r l e, Injection de semi-groupes divisibles dans des alg`ebres de convolution et construction d’homomorphismes discontinus de C(K), Proc. London Math.

Soc. (3) 36 (1978), 59–85.

[Es;2] —, A complex-variable proof of the Wiener tauberian theorem, Ann. Inst.

Fourier (Grenoble) 30 (1980), no. 2, 91–96.

[Es;3] J. E s t e r l e, Elements for a classification of commutative radical Banach alge- bras, in: Proc. Conf. on Radical Banach Algebras and Automatic Continuity (Long Beach, 1981), J. M. Bachar et al. (eds.), Lecture Notes in Math. 975, Springer, Berlin, 1983, 4–65.

[Es;4] —, Mittag-Leffler method in the theory of Banach algebras and a new approach

to Michael’s problem, in: Banach Algebras and Several Complex Variables,

F. Greenleaf and D. Gulick (eds.), Contemp. Math. 32, Amer. Math. Soc.,

1984, 107–129.

[F–L] H. G. F e i c h t i n g e r and M. L e i n e r t, Individual factorization in Banach mod- ules, Colloq. Math. 51 (1987), 107–117.

[G–R] J. E. G a l´e and T. J. R a n s f o r d, On the growth of analytic semigroups along vertical lines, Studia Math. 138 (2000), 165–177.

[G–W] J. E. G a l´e and M. C. W h i t e, An analytic semigroup version of the Beurling–

Helson theorem , Math. Z. 225 (1997), 151–165.

[G] N. G r ø n b æ k, Power factorization in Banach modules over commutative radical Banach algebras, Math. Scand. 50 (1982), 123–134.

[G–L–R] S. L. G u l i c k, T. S. L i u and A. C. M. v a n R o o i j, Group algebra modules. II, Canad. J. Math. 19 (1967), 151–173.

[H] E. H e w i t t, The ranges of certain convolution operators, Math. Scand. 15 (1964), 147–155.

[H–R;II] E. H e w i t t and K. A. R o s s, Abstract Harmonic Analysis, Volume II, Structure and Analysis on Compact Groups, Analysis on Locally Compact Abelian Groups, 3rd printing, Springer, 1997.

[J] B. E. J o h n s o n, Continuity of centralizers on Banach algebras, J. London Math.

Soc. 41 (1966), 639–640.

[K] P. K o o s i s, Sur un th´eor`eme de Paul Cohen, C. R. Acad. Sci. Paris 259 (1964), 1380–1382.

[K–V] J. Kˇr´ıˇzk o v ´ a and P. V r b o v ´ a, A remark on a factorization theorem, Comment.

Math. Univ. Carolin. 15 (1974), 611–614.

[L] M. L e i n e r t, A commutative Banach algebra which factorizes but has no ap- proximate units, Proc. Amer. Math. Soc. 55 (1976), 345–346.

[Ou] S. I. O u z o m g i, Factorization and automatic continuity for an algebra of in- finitely differentiable functions, J. London Math. Soc. (2) 30 (1984), 265–280.

[O] J.-L. O v a e r t, Factorisation dans les alg`ebres et modules de convolution, C. R.

Acad. Sci. Paris S´er. A 265 (1967), 534–535.

[Pal] T. W. P a l m e r, Banach Algebras and the General Theory of *-Algebras, Vol- ume I, Algebras and Banach Algebras, Encyclopedia Math. Appl. 49, Cam- bridge Univ. Press, 1994.

[Pas] W. L. P a s c h k e, A factorable Banach algebra without bounded approximate unit, Pacific J. Math. 46 (1973), 249–251.

[P–V] H. P e t z e l t o v ´ a and P. V r b o v ´ a, Factorization in the algebra of rapidly decreas- ing functions on R n , Comment. Math. Univ. Carolin. 19 (1978), 489–499.

[P–P] F. A. P o t r a and V. P t ´ a k, Nondiscrete Induction and Iterative Processes, Res.

Notes in Math. 103, Pitman, 1984.

[P;1] V. P t ´ a k, Un th´eor`eme de factorisation, C. R. Acad. Sci. Paris S´er. A 275 (1972), 1297–1299.

[P;2] —, Deux th´eor`emes de factorisation, ibid. 278 (1974), 1091–1094.

[P;3] —, Factorization in Banach algebras, Studia Math. 65 (1979), 279–285.

[Ri] M. R i e f f e l, On the continuity of certain intertwining operators, centralizers, and positive linear functionals, Proc. Amer. Math. Soc. 20 (1969), 455–457.

[R;1] W. R u d i n, Factorization in the group algebra of the real line, Proc. Nat. Acad.

Sci. U.S.A. 43 (1957), 339–340.

[R;2] —, Representation of functions by convolutions, J. Math. Mech. 7 (1958), 103–115.

[S] R. S a l e m, Sur les transformations des s´eries de Fourier, Fund. Math. 33 (1939), 108–114.

[S–T] F. D. S e n t i l l e s and D. C. T a y l o r, Factorization in Banach algebras and the

general strict topology , Trans. Amer. Math. Soc. 142 (1969), 141–152.

[S;1] A. M. S i n c l a i r, Bounded approximate identities, factorization, and a convolu- tion algebra, J. Funct. Anal. 29 (1978), 308–318.

[S;2] —, Cohen’s factorization method using an algebra of analytic functions, Proc.

London Math. Soc. 39 (1979), 451–468.

[S;3] —, Cohen elements in Banach algebras, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 55–70.

[S;4] —, Continuous Semigroups in Banach Algebras, London Math. Soc. Lecture Note Ser. 63, Cambridge Univ. Press, 1982.

[M.K.S] M. K. S u m m e r s, Factorization in Fr´echet modules, J. London Math. Soc. (2) 5 (1972), 243–248.

[W.H.S] W. H. S u m m e r s, Factorization in Fr´echet spaces, Studia Math. 39 (1971), 209–216.

[T] D. C. T a y l o r, A characterization of Banach algebras with approximate unit, Bull. Amer. Math. Soc. 74 (1968), 761–766.

[V] N. T. V a r o p o u l o s, Sur les formes positives d’une alg`ebre de Banach, C. R.

Acad. Sci. Paris S´er. A 258 (1964), 2465–2467.

[Vo;1] J. V o i g t, Factorization in some Fr´echet algebras of differentiable functions, Studia Math. 78 (1984), 333–347.

[Vo;2] —, Factorization in Fr´echet algebras, J. London Math. Soc. (2) 29 (1984), 147–152.

[W] M. C. W h i t e, Strong Wedderburn decompositions of Banach algebras contain- ing analytic semigroups , J. London Math. Soc. (2) 49 (1994), 331–342.

[Z;I] A. Z y g m u n d, Trigonometric Series, Vol. I, 2nd ed., Cambridge Univ. Press, 1959.

[ ˙Z] W. ˙Z e l a z k o, Banach Algebras, Elsevier, Amsterdam, and PWN–Polish Sci.

Publ., Warszawa, 1973.

Faculty of Electrical Engineering Technical University of Lublin Nadbystrzycka 38A, P.O. Box 189 20-618 Lublin, Poland

E-mail: kisynski.wzipt@antenor.pol.lublin.pl

Re¸ cu par la R´ edaction le 28.6.2000

R´ evis´ e le 13.11.2000